If the $4^{t h}, 10^{t h}$ and $16^{t h}$ terms of a G.P. are $x, y$ and $z$ , respectively. Prove that $x, y, z$ are in G.P.

Question

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Answer 1

Let $a$ be the first term and $r$ be the common ratio of the G.P.
According to the given condition,
$a_{4} = a r^{3} = x . . . (1) a_{10} = a r^{9} = y . . . (2) a_{16} = a r^{15} = z . . . (3) Dividing (2) by (1), we obtain \frac{y}{x} = \frac{a r ^{2}}{r ^{3}} \Rightarrow \frac{y}{x} = r^{6} Dividing (3) by (2), we obtain \frac{z}{y} = \frac{a r ^{15}}{a r ^{9}} \Rightarrow \frac{z}{y} = r^{6} ∴ \frac{y}{x} = \frac{z}{y} Thus, x, y, z are in G.P.$

Answer 2

Let

a

be the first term and

r

be the common ratio of the G.P.
According to the given condition,

a_{4} = a r^{3} = x . . . (1) a_{10} = a r^{9} = y . . . (2) a_{16} = a r^{15} = z . . . (3) Dividing (2) by (1), we obtain \frac{y}{x} = \frac{a r ^{2}}{r ^{3}} \Rightarrow \frac{y}{x} = r^{6} Dividing (3) by (2), we obtain \frac{z}{y} = \frac{a r ^{15}}{a r ^{9}} \Rightarrow \frac{z}{y} = r^{6} ∴ \frac{y}{x} = \frac{z}{y} Thus, x, y, z are in G.P.